Stochastic population dynamics applied to ovarian follicles development

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Stochastic population dynamics applied to ovarian follicles development

Romain Yvinec

To cite this version:

Romain Yvinec. Stochastic population dynamics applied to ovarian follicles development. BioHasard

2019, Aug 2019, Rennes, France. �hal-03115063�

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Stochastic population dynamics applied to ovarian follicles development

Romain Yvinec

Physiologie de la Reproduction et des Comportements INRA Tours

(3)

Acknowledgements

? INRIA Saclay : Fr´ ed´ erique Cl´ ement, Fr´ ed´ erique Robin

? INRA PRC : Team BIOS, BINGO (D.

Monniaux, V. Cadoret, R. Dalbies-Tran)

? INRA LPGP (J. Bobe, V. Thermes)

? INERIS (R. Beaudouin)

? C´ eline Bonnet (CMAP, X),

Kerloum Chahour (U. Cˆ ote d’Azur)

? INRA phase (Cr´ edit incitatif)

(4)

Population dynamics and ovarian follicles development

Ovarian folliculogenesis : complex multiscale dynamics

Encoding and decoding neuro- hormonal signals

Population dynamics : gametogenesis

Intra-cellular level : signaling net- works

Yvinec et al.,Advances in computational modeling approaches of pituitary gonadotropin signaling, Expert Opinion on Drug Discovery, 2018.

(5)

Gametogenesis : Ovarian folliculogenesis

• Morphogenesis and maturation of ovarian follicles

somaticandgerm(egg) cells

⇒Somatic cell division and germ cell growth up to ovulation

Gougeon & Chainy, J. Reprod. Fert. 1987

(6)

Gametogenesis : Ovarian folliculogenesis

• Pool of Quiescent follicles static reserve(perinatal in most mammals)

Slow activation

Monniaux, Theriogenology 2016

(7)

Gametogenesis : Ovarian folliculogenesis

Slow activation

• Basal growth

Dynamic reserve(starting at birth) Spanning over several ovarian cycles

(8)

Gametogenesis : Ovarian folliculogenesis

Slow activation

• Basal growth

• Terminal growth

After puberty :ovulationwithin an ovarian cycle

(9)

Gametogenesis : Ovarian folliculogenesis

Slow activation

• Basal growth

• Terminal growth

After puberty :ovulationwithin an ovarian cycle

• Interactions between all follicles via complex (neuro-) hormonal signals

(10)

Order of magnitude

Follicle population in women

• Quiescent follicles

peri-natal ≈5·10⁶ At birth ≈1·10⁶ At puberty 10⁴−10⁶ At menopause <10³

Activation rate ”A few per days”

Scaramuzzi et al., Reprod.Fert. Dev. 2011

(11)

Order of magnitude

• Growing follicles

Maturation time 120−180j Basal follicles 10³−10⁴ Terminal follicles 10² Pre-Ovulatory follicles a few

Atresia Most of them !

(12)

Order of magnitude

• Growing follicles

Maturation time 120−180j Basal follicles 10³−10⁴ Terminal follicles 10² Pre-Ovulatory follicles a few

Atresia Most of them !

-> Only 400 follicles will ever reach

(13)

Order of magnitude

• a single follicle (in women) at different maturation stages

ovocyte (egg cell) diam. : 0.01−0.1mm

follicle diam. 0.03−20mm

somatic cells diam. ≈0.01mm nb somatic cells 10²−10⁷

primary

follicle transi0onal primary

to secondary follicle small secondary

follicles large secondary follicle Oocyte growth

Granulosa cell prolifera0on

Theca and antrum forma0on ^oocyte

granulosa theca

antrum

ter0ary (antral) follicle

Courtesy of Danielle Monniaux.

(14)

Scientific and societal issues

Understanding of a complex process of developmental biology, occuring during the whole lifespan

Numerous cell types involved, and various interactions Many different spatial and temporal scales

Hormonal feedback (endocrine, paracrine, autocrine) Steric and biophysical constraint

Preserve the reproductive ability Iatrogenic or physiological alterations Sensibility to environmental conditions Biodiversity preservation

Control of the reproduction function(in humans and animals) Biotechnology of reproduction (in vivo,ex vivo,in vitro) Clinical, economical and environmental issues

(15)

Modeling ovarian folliculogenesis

Growth of a single follicle

Monniaux et al., M/S. 1999

• Thesis of F. Robin, (co- supervised. F. Cl´ ement)

Populations of follicules

Scaramuzzi et al., Reprod. Fert. Dev. 2011

• CEMRACS 2018 (summer

school), C. Bonnet, K. Cha-

hour

(16)

Follicle initiation model

(17)

Key features of follicle initiation

• Leave the quiescent phase (static reserve)

• A single layer of somatic cells

• Two types of cells : Flattened and Cuboid

• Irreversible transition from Flattened to Cuboid cells

• The follicle is ”activated”

when all cells have

transitioned

Gougeon & Chainy, J. Reprod. Fert. 1987

(18)

Stochastic model (CTMC)

, → Two cell populations : F (flattened) and C (cuboid)

, → Small number of cells : stochastic model with ponctual event, with density dependent rates.

, → Initial Condition : F = F

0

(parameter), C = 0 ; Condition finale : F = 0, C (F = 0) ≥ F

₀

(output)

Events Reaction Intensity function differentiation F→C αF+β_F^FC_+C

proliferation C→C+C γC

(19)

Stochastic model (CTMC)

, → Two cell populations : F (flattened) and C (cuboid)

, → Small number of cells : stochastic model with ponctual event, with density dependent rates.

, → Initial Condition : F = F

₀

(parameter), C = 0 ; Condition finale : F = 0, C (F = 0) ≥ F

0

(output)

Events Reaction Intensity function differentiation F→C αF+ β_F^FC_+C

,→ Retro-action of cuboid cells on the differentiation rate : is it relevant ?

(20)

Stochastic model (CTMC)

, → Two cell populations : F (flattened) and C (cuboid)

, → Small number of cells : stochastic model with ponctual event, with density dependent rates.

, → Initial Condition : F = F

₀

(parameter), C = 0 ; Condition finale : F = 0, C (F = 0) ≥ F

0

(output)

Events Reaction Intensity function differentiation F→C αF+β_F^FC_+C

,→ Retro-action of cuboid cells on the differentiation rate : is it relevant ?

(21)

Ex vivo data (snapshot data)

• Ex vivo data in sheep fetus ( Courtesy of K.

McNatty) : WT (++) vs Mutant (BB)

⇒ Proportion of cuboid cells

p

_C

= C /(F + C ) vs

number of cuboid

cells C

(22)

Ex vivo data (snapshot data)

• Ex vivo data in sheep fetus WT vs BB

• Once activated, follicles have ”fast”

cell proliferation

⇒ Are both

differentiation et de

proliferation process

concomitant or

successive ?

(23)

Ex vivo data (snapshot data)

• Ex vivo data in sheep fetus WT vs BB

• Proportion of cuboid cells seems higher in mutant than WT, for a given number of cuboid cells.

⇒ Is it coming from a

kinetic difference ?

(24)

Ex vivo data (snapshot data)

• Ex vivo data in sheep fetus WT vs BB

• Regulatory

mechanism for this process are barely known.

⇒ Is the transition of

cell differentiation

abrupt or more

progressive ?

(25)

Events Reaction Intensity function differentiation F→C αF+β FC

F+C prolif´eration C→C+C γC

• Theoretical study

⇒ Statistics of the ”transition” time τ to reach F = 0.

⇒ Variability of final cuboid cells ( E C

_τ

< ∞ if γ < α + β)

⇒ Impact of parameters e.g. on qualitative dynamics (progressive vs abrupt)

Robin et al.Stochastic nonlinear model for somatic cell population dynamics during ovarian follicle activation, (submitted) arXiv :1903.01316

(26)

• Theoretical study : stochastic bounds on τ and finite state projection algorithm are obtained thanks to coupling arguments with linear processes

F0

F

T ime

C

T ime

12 /30 Romain Yvinec BIOS, PRC, INRA

(27)

• Theoretical study

• Parameter calibration :

⇒ Adimensionalize parameter (α = 1, β ← β/α, γ ← γ/α)

⇒ Reformulation : cuboid cell number increase by 1 at each event -> it may be used as a ”counter” instead of physical time : P

∃t, (F (t), C (t)) = (f , c ) = P

F (c) = f

⇒ Likelihood : Q

n i=1

P

F (c

_i

) = f

_i

(28)

• Theoretical study

• Parameter calibration : lack of identifiability. Either γ << 1 and β unconstrained, or γ > 1 and β/γ >> 1

(29)

Agreement to data

0 5 10 15 20 25 30 35

C 0

5 10 15 20

F

Wild-Type

0 5 10 15 20 25 30 35

C Mutant

−8

−7

−6

−5

−4

−3

−2

−1 0 1

⇒ The model can capture both data sets

⇒ Lack of identifiability (non-conclusive on retro-action)

⇒ First differentiation, then proliferation (sligtly more

concomitant in mutant case)

(30)

Basal growth model

(31)

Key features of follicle basal growth

• Growth of a small follicle after initiation

• Spherical Symmetry

• Spatial structure of somatic cells in concentriclayers

• Joint dynamic ovocytegrowth

Somatic cellsProliferation

Courtesy of Danielle Monniaux.

(32)

Geometric model : spatial compartment in successive layers

• Spherical somatic cells (dG)

• Spherical ovocyte (dO)

• Finite number of Layers (J)

• Spherical Follicle (df) ,→df =d0+ 2JdG

Somatic cells are supposed incompressible and migrate to successive layers.

dO

V₁=4 3π(d_O

2+d_G)³−(d_O 2)³

⎡

⎣⎢

⎤

⎦⎥

Layer 1 Layer 2

(33)

Dynamical model (Multi-type Bellman-Harris Branching process)

• Ageandpositiondependent division rate (cell cycle regulated by the ovocyte)

• At division, unidirectional motioncentrifugal

• Cells areindependantbetween each other (Unlimited layer capacity)

p

_2,0⁽¹⁾

(34)

Dynamical model (Multi-type Bellman-Harris Branching process)

p

_0,2⁽¹⁾

p

_1,1⁽¹⁾

(35)

Dynamical model (Multi-type Bellman-Harris Branching process)

p

_0,2⁽²⁾

p

_1,1⁽¹⁾

p

_2,0⁽³⁾

(36)

• The geometrical model allows a simple spatial description

• The model is linear and

decomposable : exponential growth (under appropriate assumption), with a stable asymptotic spatial profile (with analytical first two moments) : there exists a unique λ > 0 such that the process Z

_t

verifies

t→∞

lim Z

t

e

^−λt

= ˆ Z (in law)

dO

V1=4 3π(dO

2+dG)³−(dO

2)³

⎡

⎣⎢

⎤

⎦⎥

V2=4 3π(dO

2+2dG)³−(dO

2)³

⎡

⎣⎢

⎤

⎦⎥ Layer 1

Layer 2

p_2,0⁽¹⁾

(37)

Data

• We have counting data of somatic cells in snapshot data, morphological data (diameter) and order of magnitude of transit times between follicle ”type”

t = 0 t = 20 t = 35

]Data points 34 10 18

Total cell number 113.89±57.76 885.75 ±380.89 2241.75±786.26

Oocyte diameter (µm) 49.31^±8.15 75.94^±10.89 88.08^±7.43

Follicle diameter (µm) 71.68±13.36 141.59 ±17.11 195.36 ±23.95

(38)

Data

• We have counting data of somatic cells in snapshot data, morphological data (diameter) and order of magnitude of transit times between follicle ”type”

t = 0 t = 20 t = 35

Oocyte diameter (µm) 49.31±8.15 75.94±10.89 88.08±7.43

⇒ Can we explain proliferation in concentric layers by a simple

model of ”division-migration”? Or do physical constraint play

important role ?

(39)

Data

• We have counting data of somatic cells in snapshot data, morphological data (diameter) and order of magnitude of transit times between follicle ”type”

t = 0 t = 20 t = 35

Oocyte diameter (µm) 49.31^±8.15 75.94^±10.89 88.08^±7.43

⇒ Can we characterize the growth rate of a follicle and spatial repartition of somatic cells ?

⇒ What is the impact of spatial position of a somatic cell on its

division rate ?

(40)

Fitting results

⇒ Exponential growth dominated by the first cell layer

(41)

Fitting results

⇒ Parameter identifiability and doubling time quantification (≈ 16 days) : Cell-cycle time % with ovocyte distance

Cl´ement et al.Analysis and Calibration of a Linear Model for Structured Cell Populations with Unidirectional Motion : Application to the Morphogenesis of Ovarian Follicles, SIAM App. math, 2019

(42)

Fitting results

⇒

Prediction of a stable spatial distribution (not observed in data)

(43)

More realistic model ?

∂u

∂t

+ div − → v u

= b(x)u(t, x) for x ∈ Ω(t) and with − → v linked to the negative gradient of the pressure, and the pressure related to the density...

Under locally constant density and spherical geometry, one have :

d

dt (r

_F

(t)vol(∂Ω(t)\∂Ω

_O

(t))) = γ Z

Ω(t)

b(x)dx + d

dt (r

_O

(t)vol(∂Ω

_O

(t)))

(44)

Terminal growth model (work in progress)

(45)

Terminal growth model (work in progress)

• Lost of spherical symmetry

• Joint Dynamic

Liquid-filled cavity formation and growth

Proliferation and differential of somatic cells

Morphogen gradient

• Morphodynamic of the Liquid-filled cavity formation ?

• Differentiation vs proliferation : which regulation ?

• Role of the Liquid-filled cavity ? primary

follicles large secondary follicle

Oocyte growth

Granulosa cell prolifera0on

Theca and antrum forma0on ^oocyte

antrum

ter0ary (antral) follicle Tertiary ( antral) follicle

(46)

Antrum growth model

• Lost of spherical symmetry

• Joint Dynamic

Liquid-filled cavity formation and growth

Proliferation and differential of somatic cells

Morphogen gradient

⇒ ”Advection-Diffusion-Reaction”

PDE.

∂φ_A

∂t +D∆φA = 0,x ∈ΩA(t),

∂u_M

∂t +div −v→MuM

= RM(uM),x∈ΩM(t),

∂u_C

∂t +div −→vCuC

= RC(uC),x∈ΩC(t), + Boundary conditions and

constitutive laws

primary

follicles large secondary follicle Oocyte growth

Granulosa cell prolifera0on

Theca and antrum forma0on ^oocyte

antrum

ter0ary (antral) follicle

(47)

Modeling ovarian folliculogenesis

Growth of a single follicle

Monniaux et al., M/S. 1999

• Thesis of F. Robin, (co- supervised. F. Cl´ ement)

Populations of follicules

• CEMRACS 2018 (summer

school), C. Bonnet, K. Cha-

hour

(48)

Modeling ovarian folliculogenesis on a lifespan timescale

• Compartment based model (CTMC)

• Non-linear interaction between follicles populations (endocrine and paracrine) via λ⁰s andµ⁰s.

• Several time scales (slow initiation, fast growth)

λ0 λ1 λ2

X0 → X1 → X2 → · · · Xd

↓ ↓ ↓ ↓

µ0 µ1 µ2 µd

preovulatory follicle(s) primary

follicles primordial

follicles antral

follicles secondary

follicles

Ini$a$on Basal development Terminal development Ovula$on

(49)

Modeling ovarian folliculogenesis on a lifespan timescale

• Compartment based model (CTMC)

• Non-linear interaction between follicles populations (endocrine and paracrine) via λ⁰s andµ⁰s.

• Several time scales (slow initiation, fast growth)

ελ0 λ1 λ2

1

εX0 → X1 → X2 → · · · Xd

↓ ↓ ↓ ↓

εµ0 µ1 µ2 µd

follicles

(50)

Modeling ovarian folliculogenesis on a lifespan timescale

⇒ Whenε→0 : (quasi-) stable distribution into maturity stages, driven by a slow deterministic decay of the static reserve (quiescent follicles)

ελ0 λ1 λ2

1

εX0 → X1 → X2 → · · · Xd

↓ ↓ ↓ ↓

εµ0 µ1 µ2 µd

Bonnet, et al.Multiscale population dynamics in reproductive biology : singular perturbation reduction in deterministic and stochastic modelspreprint version of a proceeding of CEMRACS 2018 : arXiv :1903.08555

follicles

(51)

Qualitative agreement with data

MJ.Faddy and R.G.Gosden

number 100000"

10000 "

1000 "

-

100

100000"

10000 '

1000 "

K

r 10000 •

1000 -

100 "

umber

1 0 2 ( a )

29

( b )

29

( c )

29 stage

-

34

stage

«

34

s tage

34 I

II

~ - ^

I I I

X f o l l i c l e s

39

f o i l

39

f o l

-

39 -

" g

44

ic l e s

-

9

44

h c l e s

-

44 a g e

m

-

a g e

•

K

a g e -

:

(years)49

H

(years)49

Figure 1. A mathematical model fitted to follicle counts from 43 human ovaries aged from 19 to 50 years of age. Each panel shows a spline-smoothed regression ( ) to the raw data representing pairs of ovaries (X) and the fitted model (—). Panels (a), (b) and (c) are follicle stages I, II and III respectively.

were balanced, however, because no follicles were dying at this stage.

The final column in Table I shows the total numbers of follicles leaving stage III and therefore summarizes the outcome of earlier stages of follicular growth leading towards ovulation.

7.284 (1.253)

7.284 (1.253) Figure 2. Schematic diagram showing the model of follicle dynamics in humans for adult ages up to and above 38 years of age (phases 1 and 2 respectively). Follicles leave stages I, II and III at the indicated growth and death rates (with SE) expressed as number per year per number of follicles present.

Table I. The i from 24 to 50 Age (years) 24-25 25-26 26-27 27-28 28-29 29-30 30-31 31-32 32-33 33-34 34-35 35-36 36-37 37-38 38-39 3 9 ^ 0 40-41 4 1 ^ 2 4 2 ^ 3 43-44 44-45 45-46 46-47 47-48 4 8 ^ 9 49-50

numbers of follicles moving from stage i years of age

I->

13617 12 099 10 750 9552 8487 7541 6700 5953 5290 4700 4176 3710 3297 2929 6099 4506 3329 2459 1817 1342 991 732 541 400 295 218

(predicted from model)

13617 12 099 10 750 9552 8487 7541 6700 5953 5290 4700 4176 3710 3297 2929 2381 1759 1299 960 709 524 387 286 211 156 115 85

II->

18 399 16 764 15 189 13 704 12 324 11 055 9896 8845 7896 7042 6276 5589 4975 4427 3914 3366 2822 2322 1884 1511 1200 947 742 578 448 346

to stage per

18 399 16 764 15 189 13 704 12 324 11055 9896 8845 7896 7042 6276 5589 4975 4427 3914 3366 2822 2322 1884 1511 1200 947 742 578 448 346

annum

IIl->

18 626 16 986 15 401 13 903 12 508 11 223 10 050 8984 8022 7155 6377 5680 5056 4499 3986 3442 2895 2388 1941 1559 1240 979 767 598 464 359

numbers leaving stage I. It may seem surprising that the numbers in the final column exceeded those in the first, but this is due to there being many follicles already in stages II and III at these ages, contributing to eventual egress from stage III.

Although the numbers of follicles at stage III are falling, they actually increase when expressed as a fraction of those

at INRA Institut National de la Recherche Agronomique on February 7, 2016http://humrep.oxfordjournals.org/Downloaded from

28 /30 Romain Yvinec BIOS, PRC, INRA

(52)

Summary

Population dynamics in ovarian folliculogenesis

• Somatic cells differentiation and proliferation during fol- licle initiation

• Somatic cells proliferation and migration during basal follicle growth

• Multiscale nonlinear dyna-

mics shape the follicle po-

pulation distribution into dif-

ferent maturity stages.

(53)

Summary

Population dynamics in ovarian folliculogenesis

• Somatic cells differentiation and proliferation during fol- licle initiation

• Somatic cells proliferation and migration during basal follicle growth

• Multiscale nonlinear dyna- mics shape the follicle po- pulation distribution into dif- ferent maturity stages.

Thank you for your attention !

(54)

Role of the Liquid-filled cavity ?

Redding et al,Mathematical modelling of oxygen transport-limited follicle growth, Reproduction, 2007

• Antrum formation is driven by oxygen (and nutriment) accessibility of the oocyte

• Antrum size is driven by the need to increase the number of somatic cells (without increasing the thickness of the somatic layers) to supply estrogen production in agreement with body